MIT OpenCourseWare

Browse materialsMIT 18.755 · Fall 2004

Introduction to Lie Groups

Subjects: Algebra · Geometry · Mathematics

Art: Max Brückner (1860 – 1934), Brückner Vielflache Tafel 07 · Public domain

Instructor: Prof. Sigurdur Helgason
Term: Fall 2004
Level: Graduate
What's inside: 1 lecture note · 1 reading · 13 other files · 3 web pages
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Source: MIT OpenCourseWare 18.755 Introduction to Lie Groups, Fall 2004, Prof. Sigurdur Helgason — original course page on ocw.mit.edu ↗. License: CC BY-NC-SA 4.0. Study order: MIT's own listing order, section by section. Taken from MIT's published course files, downloaded from ocw.mit.edu on September 23, 2026. How RedLectern is built.

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MIT OpenCourseWare, Prof. Sigurdur Helgason, 18.755 Introduction to Lie Groups, Fall 2004.

License: CC BY-NC-SA 4.0 · Edition: Fall 2004 · Open the original course on OCW ↗

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Introduction to Lie Groups

Before you start

What's here, and what isn't

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Materials by section

Readings

Lecture Notes

Assignments

Lecture 1: Historical Background and Informal Introduction to Lie Theory

Lecture 2: Differentiable Manifolds, Differentiable Functions, Vector Fields, Tangent Spaces

Lecture 3: Tangent Spaces; Mappings and Coordinate Representation Submanifolds

Lecture 4: Affine Connections Parallelism; Geodesics Covariant Derivative

Lecture 5: Normal Coordinates Exponential Mapping

Lecture 6: Definition of Lie Groups Left-invariant Vector Fields Lie Algebras Universal Enveloping Algebra

Lecture 7: Left-invariant Affine Connections The Exponential Mapping Taylor’s Formula in a Lie Group Formulation The Group GL (n, R )

Lecture 8: Further Analysis of the Universal Enveloping Algebra Explicit Construction of a Lie Group (locally) from its Lie Algebra Exponentials and Brackets

Lecture 9: Lie Subgroups and Lie Subalgebras Closer Subgroups

Lecture 10: Lie Algebras of some Classical Groups Closed Subgroups and Topological Lie Subgroups

Lecture 11: Lie Transformation Groups A Proof of Lie’s Theorem

Lecture 12: Homogeneous Spaces as Manifolds The Adjoint Group and the Adjoint Representation

Lecture 13: Examples Homomorphisms and their Kernels and Ranges

Lecture 14: Examples Non-Euclidean Geometry The Associated Lie Groups of Su (1, 1) and Interpretation of the Corresponding Coset Spaces

Lecture 15: The Killing Form Semisimple Lie Groups

Lecture 16: Compact Semisimple Lie Groups Weyl’s Theorem proved using Riemannian Geometry

Lecture 17: The Universal Covering Group

Lecture 18: Semi-direct Products The Automorphism Group as a Lie Group

Lecture 19: Solvable Lie Algebras The Levi Decomposition Global Construction of a Lie Group with a given Lie Algebra

Lecture 20: Differential 1-forms The Tensor Algebra and the Exterior Algebra

Lecture 21: Exterior Differentiation and Effect of Mappings Cartan’s Proof of Lie’s Third Theorem

Lecture 22: Integration of Forms Haar Measure and Invariant Integration on Homogeneous Spaces

Lecture 23: Maurer-Cartan Forms The Haar Measure in Canonical Coordinates

Lecture 24: Invariant Forms and Harmonic Forms Hodge’s Theorem

Lecture 25: Real Forms Compact Real Forms, Construction and Significance

Lecture 26: The Classical Groups and the Classification of Simple Lie Algebras, Real and Complex